In total, we have planed 4 mathematic models including epidemiological analysis of breast cancer, glucose metabolism, and lactate production that are designed in 2022, as well as adaptative metabolic and proteomic network evolution of bacteria living on lactate that will be designed in 2023.
There are millions of people diagnosed with breast cancer globally. A mathematic model is developed for the epidemiological determination of factors that may be associated with the occurrence and prognosis of breast cancer within a population. It is aimed for the disease prevention and early detection. Preliminary results generated by the model may be applied to inform public health practices and mitigate risk. We have launched an online epidemiological questionnaire that was open and publicly available (Figure 1). In total, we have received 712 valid questionnaires between February 18 and July 10, 2022. These valid questionnaires cover 20 provinces or regions of China.
Figure 1. Epidemiological questionnaire on breast cancer
Breast cancer is a disease characterized by the growth of malignant cells in mammary glands.1 According to the data gathered by International Agency for Research on Cancer, breast cancer takes up 11.7% of the new cancer incidence, and 6.9% of the deaths in 2020.2 Breast cancer affects men and women worldwide.3 Although some risks have been assessed, many are still under investigation.4-6
One of the most common symptoms of breast cancer is the formation of lumps, although most lumps or masses are benign and not cancerous in the breast.7 Other possible early symptoms appear in both the look and feel of the breast that are summarized as follows.8-11 Lots of efforts continuously focus on the development of various techniques that can promote more detailed electronic health records for early self-judgement with machine learning, and automatic recognition.12-16
- Skin, such as dimpled, puckering or scaly skin.
- Size, shape, skin texture or color.
- Nipple discharge other than breast milk.
- Nipple appearance, such as inversion, tenderness or flaking.
- Swelling or enlarged lymph nodes in the armpit.
Breast cancer is not considered as an infectious disease17 because of the absence of evidences for viral or bacterial infections.18-20 The occurrence of breast cancer is due to the accumulation of genetic mutations such as BRCA1, BRCA2, TP53, CDH1, and PTEN in mammary gland epithelial cells.5,21 Those encoded proteins play important roles in the tumorigenesis of breast cancer.22-24 High risks of occurrences is often associated with the family history of breast cancer.4, 5, 25 It should be noted that many tumors do not cause visual symptoms that can be seen or felt in early stages. Advanced techniques have been developed for early screening and accurate diagnosis including mammography, MRI, ultrasound and minimally invasive biopsy.26 Early detection by regular checking of breast changes helps to take less invasive treatment as soon as possible and brings about better outcome. Affordable small-scale devices have emerged for individual’s daily health care through the measurement of impedance, surface tension, local temperature, or absorption of irradiations in different wavelengths.27–37 Advancement in analytical techniques has promoted more accurate diagnosis and precise oncologic decisions based on molecular classification in addition to conventional clinicopathologic features or routine biomarkers.38-42 The dissection of genetic, proteomic and metabolic alterations brings new insights into treatment strategies and management of breast cancer.44-47
Here we adopt the categorization method in the preliminary epidemiological survey that focuses on symptoms appearing in the early stage of the disease.4–6, 25, 48 Age, gender, educational level, family history of related illness, behavioral and reproductive factors have been taken into the consideration. Follow-up investigations may be conducted in the future in order to validate the risk factors of breast cancer.50-52 Table 1 shows the cross-sectional design53 of the preliminary questionnaire for the gathering and analysis of relevant data in order to investigate the current prevalence of breast cancer in subsets of the population.
| Table 1. Design of the questionnaire | |
|---|---|
| Category | Sub-category |
| Age | |
| Gender | |
| Education background | |
| Family history | |
| Alcohol consumption | |
| Diet | |
| Behavioral factors | Cigarette |
| Stay up | |
| Physical exercise | |
| Number of giving birth | |
| Breast feed | |
| Reproductive factors | Termination of pregnancy |
| Menarche time | |
| Menstrual cycle | |
The threshold for the existence of potential strong association measured by the phi coefficient is set at 0.7 and -0.7.54 To determine the association between factors and the disease, we first examine the relationship among factors. Since our questionnaire are designed as binary questions, for example, whether suffers from breast cancer, we adopt the logistic model to demonstrate the associations. Note that not all of the questions are binary, we discuss the binary questions and non-binary questions separately.55
We here assume the total number of the respondents to be for generality. For the -th respondent, we define the random variables as shown in Table 2 and two generalized random vectors and .
| Table 2. Variables used in mathematic modeling | |
|---|---|
| Variables | Definition |
| Yi | 0 for not suffering from breast cancer, 1 for otherwise |
| Xi1 | 0 for male, 1 for otherwise |
| Xi2 | 0 for education background lower than college or undergraduate, 1 for otherwise |
| Xi3 | 0 for none of the family members have ever suffered from the disease, 1 for otherwise |
| Xi4 | 0 for never alcohol drinker, 1 for otherwise |
| Xi5 | 0 for never smoker, 1 for otherwise |
| Xi6 | 0 for none or seldom physical exercise, 1 for otherwise |
| Xi7 | 0 for having an unhealthy diet, 1 for otherwise |
| Xi8 | 0 for possessing the habit of staying up, 1 for otherwise |
| Ui1 | 0 for younger than 17, 1 for 18-29, 2 for 30-39, 3 for 40-49, 4 for older than 50 |
| Ui2 | 0 to 3 is equal to the time the respondent had given birth, 4 for more |
| Ui3 | 0 for doesn’t feed by breast, 1 for less than 12 months, 2 for 13 to 24 months, 3, for 25 to 36 months, 4 for more |
| Ui3 | 0 to 2 is equal to the time the respondent had terminated pregnancy, 3 for more |
| Ui4 | 0 to 3 is equal to the time the respondent had given birth, 4 for more |
Because only binary random variables are involved, we deal with the variables in an iterative sequence in order to study the relationships among random variable pairs. For each pair consisting of 2 random variables, for instance, and , there are merely 4 possible cases, we label them in alphabetical order as shown in Table 3.
| Table 3. Relationship among random variable pairs | ||
|---|---|---|
| Xi\Xj | 0 | 1 |
| 0 | Case A | Case B |
| 1 | Case C | Case D |
Binary variables here are symmetrical in significance. In other words, each value of the variable is of equal weight when determining the association. To measure the association between two binary variables, we use the phi coefficient introduced by Udny Yule.56 Phi correlation coefficient is a metric used to determine the correlation between two binary variables, which is a special case for the Pierson coefficient.57 It is widely used in the field of bioinformatics.58 The range of phi coefficient is . The closer the phi coefficient to 1, the stronger positive correlation exhibits. Similarly, the closer the phi coefficient to -1, the stronger the negative correlation exhibits. The phi coefficient closer to 0 indicates weak or no correlation. The number of respondents in each case is denoted as as shown in equation (2) and Table 4.
| Table 4. Definition of phi coefficients. | |||
|---|---|---|---|
| i \ j | 0 | 1 | Total |
| 0 | nA | nB | n0· |
| 1 | nC | nD | n1· |
| Total | n.0 | n.1 | N |
The formula for the phi coefficient is defined as
Through iterative computation, we obtain the result shown in Figure 2.
Figure 2. Plot of the phi coefficients with random variables.
Because all of the phi coefficients are below 0.5, we can regard binary variables as linearly irrelevant. Interestingly, the habit of staying up and having an unhealthy diet produces the largest positive phi coefficient, indicating those people who have the habit of staying up tend to have an unhealthier diet. The second-largest phi coefficient is associated with smoking and drinking. It means those people who possess the habit of drinking alcohol are also likely to smoke cigarettes. Moreover, the random variables and show a negative correlation with a phi coefficient equal to -0.281, indicating men are more likely to have the habit of smoking cigarettes than women. The phi coefficients of the other random variable pairs are less than 0.2 in absolute value.
To elucidate the association between binary random variables and non-binary random variables, we first notice that some random variables are already correlated biologically, for instance, gender () and reproductive factors. We seek to identify those correlated random variable pairs in the first place and label them as BC (biologically correlated). The BC pairs are shown in Table 5.
| Table 5. Biological correlations of non-binary variables. | |
|---|---|
| BC Pair | Justification |
| (X1,U2) | Only female has the possibility to bear children |
| (X1,U3) | Only female has the possibility to feed by breast |
| (X1,U4) | Only female has the possibility to terminate a pregnancy |
| (X1,U5) | Only female has the possibility to have menstrual |
| (U2,U3) | Only a female given birth can she has the possibility to feed by breast |
The mathematical tool we are going to adopt here is the coefficient introduced by M. Baak, R. Koopman, H. Snoek, and S. Klous, which serves as an extended and amended measure of Pearson’s correlation coefficient. As constructed, Pearson’s correlation coefficient is only applicable when studying interval variables.59 Moreover, along with Pearson’s correlation coefficient, many statistical measures can only capture linear correlation, which would produce enormous errors and false interpretations when the association is not linear.54, 60 The variables we are studying here are categorical. Some coefficients, for instance, Cramr’s V, are specifically designed to measure the correlation between categorical variables.59 However, their values are dependent on the number of rows and columns of the contingency table.60
There are rows and columns are in the contingency table and represents a cell in the contingency table. The cells are referred as . The total case number is defined as . For each random variable pair denoted by and , their probability distribution function is defined as formular (4).
Herein, x, y denote the average value and σx,σy stand for the standard deviation, respectively.The ρ represents the Pearson correlation coefficient. We denote the observed value and the expected value of cell (i,j) by Oi,jEi,j respectively. The formula of Pearson’s χ2 test is defined as formula (5).
Using the definition of the probability density function, the probability of cell is defined as
By assigning the value
Substitute to Formula (5), we obtain
It is noted the observed value is constructed independently on true observed values. We then introduce the effective number of freedom and the number of observed empty cells to treat statistical noises.
We define the pedestal statistics χp2 and χmax2 as follows.59
Where c is added to exclude the outliers. To amend the Formula (8) by extending for case ρ=0 and case ρ = 1, we obtain the formula (12).
The correlation coefficient φk can be calculated as follow.
The ρ` can be obtained using Brent's method.14 Note that χ2 p and χ2 max(N,r,k) are real values dependent on r,k and N. After the calculation with SPSS and Python, we obtain the result shown in Figure 3. The code we used can be found in the reference.61
Figure 3. Correlations coefficients generated with Matplotlib.
The number in each cell indicates the φk coefficient of the corresponding random variable pair. The closer φk to 1, the tighter correlation the random variable pair exhibits. On the contrary, the closer φk to 0, the more independence the random variable pair exhibits. Some biologically associated factors such as gender (X_1) and the number of giving birth (U_2) are adjusted in order to obtain a valid correlation coefficient.
It can be found in Figure 3 that random variable pairs exhibit strong correlation include (X7,X8),(U1,U3) , and (U1,U2) ,excluding the biologically correlated factors. As we just demonstrated, the first random variable pair indicates that those with the habit of staying up tend to have an unhealthy diet. The second pair indicates that the breastfeeding time is correlated with the age of the respondent with φk=0.53. The third pair indicates that the number of giving birth is correlated with the age of the respondent with φk=0.81, which shows a strong correlation.
It is noted that all these correlated variables are categorized in the reproductive factors. We plan to establish a combined random variable that can represent the overall effects in order to satisfy the linear independence requirement of the logistic regression in the next section. The method of principal component analysis is applied to integrate the associated random variables. It is commonly used in exploratory data analysis and able to conduct dimensional reduction with a vector projection.62 The principal components are exactly the eigenvectors of the covariance matrix of acquired data, which is obtained in previous procedures. The covariance matrix of U1,U2,and U3 is shown below.
By calculating the eigenvalues and related eigenvectors with the SVD decomposition of C0, we obtain the adjusted coordinate system.63 Then the greatest variance lies on the first principal component, with the second greatest variance on the second principal component and so on. Figure 4 shows the eigenvalues of each random variables.
Figure 4. Eigenvalues of each random variables.
Since the eigenvalues of last two variables are less than 1, we perform the dimensional reduction by decorrelating U2,U3 .64 The feature vector is denoted by Ν shown in the formula (15).
Through the analysis above, the independent factors are identified. Considering the interaction among correlated factors, we combine their effect into an overall effect-equivalent random variable. We define a new set of variables in Table 6.
| Table 6. Variables and assigned values | |
|---|---|
| Variables | Definition |
| Vi1 | 0 for male, 1 for otherwise |
| Vi2 | 0 for education background lower than college or undergraduate, 1 for otherwise |
| Vi3 | 0 for none of the family members have ever suffered from the disease, 1 for otherwise |
| Vi4 | 0 for never alcohol drinker, 1 for otherwise |
| Vi5 | 0 for never smoker, 1 for otherwise |
| Vi6 | 0 for none or seldom physical exercise, 1 for otherwise |
| Vi7 | 0 for having an unhealthy diet, 1 for otherwise |
| Vi8 | 0 for possessing the habit of staying up, 1 for otherwise |
| Vi9 | The integrated random variable of Ui1,Ui2,Ui3 |
| Vi10 | 0 to 2 is equal to the time the respondent had terminated pregnancy, 3 for more |
| Vi11 | 0 for never had menarche, 1 for younger than 11, 2 for 11 to 14, 3 for 15-18, 4 for later than 18 |
The logistic regression method is applied to model the relationship between breast cancer status and the linear combination of potential factors we have established.
Define D={1,2···,M},E={0,1,···,M},then
Where p is the probability for an individual to have breast cancer and {Vk} is the set of potential factors, and βk is the weight for the -th potential factor respectively. Using logit transformation, we convert the non-linear relationship between p and Vk into a linear relationship between logit(p) and Vk.
To obtain the most likely value of βi, we use the maximum likelihood method by
Where yk is the observed breast cancer variable for the k-th individual. Taking the partial derivative for each βi and assigning them to 0, we obtain the following set of equations, in which Vnk represents the value of Vn of k-th individual.
The result of {βi} is shown in Table 7. Bradford Hill criteria are used in studies of the association and causality between environmental factors and the disease.65 It’s widely accepted in determining the causal relationship between factors and the disease.66 The sequential order is based on the weight of each factor as shown in Formula (16). When β increases, the probability of having breast cancer p increases.
| Table 7. Coefficients of the occurrence of breast cancer | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Coefficient | β0 | β1 | β2 | β3 | β4 | β5 | β6 | β7 | β8 | β9 | β10 | β11 |
| Values | 0.000 | 3.179 | 0.361 | -10.016 | 2.340 | 19.945 | -1.100 | 19.945 | 22.893 | 15.781 | 1.999 | -2.233 |
| Sequence | NA | 10 | 7 | 11 | 5 | 2 | 8 | 3 | 1 | 4 | 6 | 9 |
It has been noted that limited sampling population strongly affect the accuracy of the results. For example, the coefficient of education background (V2) is calculated as 0.361, which means the higher the education background is associated with the higher risk of breast cancer. The biased result is due to the fact that the majority of our questionnaire was collected from undergraduate students in universities. Such biased sampling increases the probability that a respondent has a high education background. Then educational background cannot be defined as the deterministic factor that is associated with the occurrence of breast cancer. Another example is the coefficient for gender (V1) that is -3.179, meaning males are more likely to have breast cancer. It is also obviously biased25 ,49 resulting from the fact that female respondents takes up the majority of collected questionnaire. Additionally, because the majority of our respondents are healthy undergraduate students, the coefficient of menarche time (V11) is biasedly reduced.
Staying up late may reduce the production of melatonin that may suppress the proliferation of cancer cells. It increases the occurrence risk of breast cancer.67 This effect is amplified especially in the group consisting of those who are deprived of sleep such as night shift workers.68,69 The exposure to light may cause the downregulation of melatonin.70
Unhealthy diet usually refers to the food containing high content of sugar, fat, and salt.71 It has been considered as a major reason causing obesity that is tightly correlated to breast cancer. In particular for those women who pass their menopause, the obese group has a higher risk of breast cancer.72 However, obesity occurring in childhood and adolescence somehow drops this risk with unknown underlying mechanism.73 By having a healthy diet such as vegetables and fruits, the possibility of the occurrence of breast cancer in middle age is significantly lowered.74
As suggested by other breast cancer epidemiological surveys, the occurrence risk of breast cancer is significantly higher in the smoking group.75 It was found that those people who have a family history of smoking have a higher risk.76 Nicotine is significantly correlated with lung metastases of breast cancer. The carcinogens present in the cigarette are founded in breast tissues.77
Drinking alcohol is an indirect cause of multiple malignant tumors. But it affects hormone levels prominently.78 By reducing alcohol intake, a variety of diseases can be avoided, including oropharyngeal cancer, throat cancer, and esophageal cancer. The intake of alcohol is positively correlated with the risk of having breast cancer.79 By limiting alcohol intake, we can significantly reduce the occurrence risk of breast cancer.
The overall reproductive factor that combines age, number of giving birth, menarche, menopause and breastfeeding time. The underlying core of the reproductive factors is the change in hormone levels.80 The lag of 2 years of menarche corresponds to a 10% lower occurrence risk of breast cancer.72 The difference in menopause time corresponds to a 17% difference in breast cancer risk.72,81 Moreover, lifelong infertility and late pregnancy increase the occurrence risk of breast cancer.81
Full pregnancy reduces the occurrence risk of breast cancer. However, the effect of incomplete pregnancy on the occurrence of breast cancer remains unknown.82 In some epidemiological research, neither the times of pregnancy termination nor the pregnancy period shows a significant correlation with the occurrence of breast cancer, which is not in accordance with our result.82,83 More relevant investigation is required to draw a conclusion.
We have used the Kaiser-Meyer-Olkin test and Bartlett’s test to assess the sample adequacy and the null hypothesis, respectively.84,85 Applying Formula (20) below, we obtain the sample’s KMO index that is 587. In the formula,rjk represents the correlation between the desired variable pair and pjk represents the partial correlation. Since our KMO index falls in the interval , our result is acceptable.86 In Formula (21 ) ,
and
represent the pooled estimate for the variance. Because the approximate χ2 of our sample is 42.262 and the significance level is 0.000, our conclusions are reasonable.
Moreover, we have applied the log-likelihood test and the degree of freedom test to evaluate the significance of our data. The detail of the test result can be found in Appendix B. Table 8 lists the estimation accuracy of our model. Although in some cases our model exhibits low estimation accuracy, the overall accuracy is 93.50%.
| Table 8. Evaluation of the epidemiological model | |||
|---|---|---|---|
| Real Value | Evaluation | Accuracy | |
| 0 | 1 | ||
| 0 | 397 | 9 | 97.80% |
| 1 | 21 | 32 | 60.40% |
| Accuracy | 91.10% | 8.90% | 93.50% |
We have analyzed the associations between early symptoms and the occurrence of breast cancer listed in Table 9
| Table 9. Correlation of early symptoms and the occurrence of breast cancer | |
|---|---|
| Number | Symphtom |
| 1 | 0 for malA lump or thickening in/near the breast or in the underarm area |
| 2 | 0 for educaNipple discharge |
| 3 | Irregular menstruation, anxious or depressed |
| 4 | Local skin hyperthermia, redness, skin abscess |
| 5 | Swollen lymph nodes |
| 6 | Dimpling of the Breast tissue, orange peel alike |
| 7 | Inverted nipples |
Those symptoms are selected because they are distinctive and common for breast cancer patients.8–10 As Figure 5 shows below, most of the patients have a lump or thickening in/near the breast or in the underarm area. The second largest group has irregular menstruation, anxious or depressed feelings. Other symptoms are nipple discharge, inverted nipples, local skin hyperthermia, redness, and skin abscess.
Figure 5. Correlation of early symptoms with the occurrence of breast cancer
Question 18 and 19 on our questionnaire investigates the diagnostic technique and treatment methods, respectively. According to the data collected from the questionnaire as shown in Figure 6, the majority of our respondents selected general breast inspection (38%), followed by breast mammogram (22%), molybdenum target mammogram (14%), tissue biopsy inspection (10%), and blood test (6%). Due to our limited sample size, we only consider those higher than 5% as frequently chosen methods.
Figure 6. Currently used diagnostic techniques.
We describe the frequency of used treatment method by both absolute and relative measurements to avoid unintended intuition from data as shown in Table 10.
| Table 10. Medical treatments | ||
|---|---|---|
| Number | Treatment | Notation |
| 1 | Drugs | Ti1 |
| 2 | Resection operation | Ti2 |
| 3 | Laser therapy | Ti3 |
| 4 | Chemotherapy | Ti4 |
| 5 | Radiotherapy | Ti5 |
| 6 | Endocrine therapy | Ti6 |
| 7 | Targeted therapy | Ti7 |
| 8 | Self-healing | Ti8 |
Since there are multiple choices of treatment methods, we simplify the cases by dividing each respondent’s choice into multiple binary choices. Ci represents the cure status of the i-th respondent, in which 1 for cured, 0 for still suffering from the disease. We measure the associations between cure status and the treatment measures as shown in Table 11.
| Table 7. Coefficients of the occurrence of breast cancer | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Treatment | T1 | T1 | T1 | T1 | T1 | T1 | T1 | T1 | ||||
| Select count | 26 | 11 | 0 | 1 | 1 | 2 | 0 | 18 | ||||
| Phi coefficient | 0.138 | 0.506 | None | -0.078 | 0.137 | 0.000 | None | -0.236 | ||||
It is noted that the 3rd and 7th treatment methods are selected by none and their phi coefficients are labeled as none. So we did not consider them in the later discussion. Also, we eliminated chemotherapy (4th), radiotherapy (5th), and endocrine therapy (6th) because their selection count is less than 5. Of all the other 3 treatment methods we surveyed, the drug is selected by the majority and the following are self-healing, then resection operation. Self-healing’s phi coefficient is -0.236, which indicates a weak negative correlation.
| Table 12. Resources for cancer research | |
|---|---|
| Institutes | Websites |
| Health Sciences Library System | https://www.hsls.pitt.edu/obrc/ |
| International Agency for Research on Cancer | https://gco.iarc.fr/projects#datavisualization |
| American Cancer Society | https://www.cancer.org/ |
| CDC | https://www.cdc.gov/cancer/breast/statistics/index.htm |
| China National Central Cancer Registry | https://ghdx.healthdata.org/series/china-national-central-cancer-registry |
| World Cancer Research Fund International | https://www.wcrf.org/cancer-trends/breast-cancer-statistics/ |
| Cancer Incidence in Five Continents | CI5 - Home (iarc.fr) |
| Physician Data Query | https://www.cancer.gov/publications/pdq |
| Table 13. Resources for treatment | |
|---|---|
| Institutes | Websites |
| National Breast Cancer Foundation | https://www.nationalbreastcancer.org/ |
| Breast Cancer Foundation | https://www.bcf.org.sg/ |
| Cancer Support Community | https://www.cancersupportcommunity.org/breast-cancer |
| Cancer Care | https://www.cancercare.org/diagnosis/breast_cancer |
| TNBC Foundation | https://tnbcfoundation.org/ |
| Breast Cancer Now | https://breastcancernow.org/ |
| NBCCEDP | https://www.cdc.gov/cancer/nbccedp/ |
| National Mammography Program | https://www.nationalbreastcancer.org/national-mammography-program |
We have conducted an epidemiological survey of breast cancer using online questionnaires. Based on the collected data, we identified two major risk factors in correlation with breast cancer including behavioral factors and reproductive factors. Behavioral factors include the habit of staying up late, smoking, unhealthy diet, and alcohol drinking. Reproductive factors include age, number of giving birth, breastfeeding time, and pregnancy termination. We have applied coefficient and amended Pearson coefficient to the determination of the relationship of deterministic factors and associated variables are combined with principal component analysis. The accuracy of our model is estimated with the logistic regression and justified with Kaiser-Meyer-Olkin test and Bartlett test. We hope make some useful contribution to our community and raise public awareness for early detection and prevention of breast cancer.
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Glucose metabolism is central to energy consumption in biological systems.1-6 Triple negative breast cancer (TNBC) tumors usually undergo reprogramming into highly glycolytic tumors through the Warburg effect, which causes the accumulation of metabolic byproduct lactate and extracellular acidification.7-11 This project is aimed to establish a mathematic model for in vitro cells that can monitor the dynamic metabolism and explore roles of lactate regulators in the production of lactate, including glucose transporters, lactate dehydrogenases, and lactate transporters. For proof-of-principle demonstration and simplification, we choose MCF 7 and MDA-MB-231cells in contrast to MCF 10A cells for the experimental validation of the established mathematic model (See Measurement and Results page for more information). Figure 7 represents the known metabolism pathways of glucose and lactate in cells.
Figure 7. The glucose and lactate metabolism in cells
The MCF 10A cell line is a non-tumorigenic epithelial cell line. It was isolated in 1984 from a mammary gland. The cells are positive for epithelial sialomucins, cytokeratins and milk fat globule antigen.12 MCF-7 is a cell line that was first isolated in 1970 from the breast tissue of a 69-year old woman. It is a widely studied epithelial cancer cell line that has characteristics of differentiated mammary epithelium.13 These cells can be easily used for detecting PI3K and MAPK, as well as ERK and Akt phosphorylation.14 They have the ability to process estradiol via cytoplasmic estrogen receptors.15 The MDA-MB-231 cell line was first isolated by M D Anderson from a pleural effusion of a patient with invasive ductal carcinoma, and the genome clusters with the basal subtype of breast cancer.16 It is commonly used to model late-stage triple negative breast cancer.17
Lactate regulators including glucose transporters, lactate dehydrogenases, and lactate transporters (MCT4) are selected as major parameters for establishing the mathematic models. The upregulation of transporters or enzymes has been observed in many different types of cancers for accelerated metabolism.18 Blocking the activity of these transporters might lead to the death of the tumor cell.19 Based on these literature studies, the initial number and the number at the time t of cancer cells are defined as N0 and N I in the following modeling. Although enhanced glucose uptake for glycolytic ATP generation or anabolic reactions is favorable for tumor growth, the abnormal tumor microenvironment is the deterministic for the adoption of metabolic phenotypes of tumor cells. Adaptative metabolic changes may result from the spatial and temporal heterogeneity in oxygenation, pH, and the concentrations of glucose, lactate, and many other metabolites of tumor microenvironments. Additionally, extracellular lactate can be recycled by oxygenated cells in tumors with another transporter (MCT1) to fuel respiration and preserve glucose for hypoxic cells. The best documented aerobic glycolysis is not a universal feature of all human cancers. Even in glycolytic tumors, oxidative phosphorylation is not completely shut down.20 All these known and unknown factors make a very complicated and concerted metabolic network for maintaining the progression of tumor cells. In this project, their accumulated effects have been considered and combined as an overall coefficients K that represents the effects of not only the three known regulators but also other unknown regulators in the complex metabolic network. K1, K2, and K3 represent the effects of three inhibitors including WZB117, Galloflavin, and AZD3965, respectively.
Define B, D, G, and t as the cytogenetic rate, cell death rate, glucose concentration, and growth time, respectively. As for the first step modeling, all factors that affect the glucose metabolism are simplified as independent parameters and their combined effect is considered as the linear addition of all factors. Then we can obtain the equation 1 that can be derived to equation 2-5. In these equations, V and represents the consumption rate dG/dt and averaged consumption rate G/t of glucose, respectively.
G represents the concentration of glucose at the time t. A liquid chromatographic technique was established for proof-of-principle demonstration of the dynamics of glucose metabolism. Samples were collected from the culture medium of cancer cells in which different quantities of different inhibitors were added. Glucose and lactate were separated from the ingredients present in the complex culture medium with a C18 reversed column. N0, Ni, G, V, and t can be obtained from the experiments and then those unknown coefficients K1, K2, K3, K and B can be determined.
For simplification, it is assumed that lactate is only produced from pyruvate. In equation 6, L represents the concentration of lactate at time t. Other unknown parameters are defined as a and K´
The equation 5 can be changed to equation 7.
Then
Changes in lactate concentration under the treatment of different inhibitors with concentrations can be experimentally measured with liquid chromatography. Those parameters such as K´ and a can also be experimentally determined.
Two preliminary mathematic models have been established for the in vitro monitoring of the dynamic changes of glucose and lactate of cancer cells. These preliminary models represent our first step towards the development of more comprehensive metabolic and proteomic networks that can explore the dynamics of glucose and lactate metabolism of cells. Coefficients described in these two models can be experimentally demonstrated with a liquid chromatographic technique. It has been used to measure dynamic changes in concentrations of glucose and lactate of cells in the culture medium under the treatment of inhibitors of glucose transporters, lactate dehydrogenases, and lactate transporters (MCT4) with different concentrations. We hope our preliminary models can be helpful for the iGEM community.
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